JournalofAgricultural and Resource Economics 24(1):204-221
Copyright 1999 Western Agricultural Economics Association
Demand for Herbicide in Corn:
An Entropy Approach Using
Micro-Level Data
Channing Arndt
Price responsiveness of herbicide demand in corn for farmers in Indiana's White
River Basin using cross-section data from individual farms is estimated. Particular
attention is paid to appropriate treatment of binding nonnegativity constraints.
Estimation was first attempted using an approach to demand systems estimation
suggested by Lee and Pitt. However, analytical and computational difficulties
effectively preclude estimation by the Lee and Pitt approach. As an alternative, a
maximum entropy (ME) approach is presented and discussed. Results from the ME
estimator tentatively indicate limited response of herbicide demand to changes in
own prices. The maximum entropy approach to demand systems estimation appears
to have merit and warrants further attention.
Key words: censored dependent variables, demand systems, herbicides, maximum
entropy, reservation prices
Introduction
Recently, environmental considerations have generated support for taxation of herbicides (Rudstrom). The incidence of taxes on specific herbicides depends largely upon
the ability of farmers to substitute other herbicides or other means of weed control.
If substitution possibilities exist, the environmental goal of reduction in use of certain
herbicides may be met and, at the same time, farmers will bear only a portion of
the incidence of the tax. On the other hand, if substitution possibilities are limited,
reductions in use of the taxed herbicides will be more limited and farmers will likely
bear the brunt of the incidence of the tax. Factor demand elasticities, derived from
demand system parameters, summarize the ability of farmers to change herbicide use
in response to price changes.
Essentially, two basic approaches may be employed in estimating demand system
parameters: (a) aggregate analysis where total consumption of goods in the system is
regressed against changes in relative prices over time, and (b) individual analysis where
consumption of goods in the system by different agents is regressed against the price
vectors faced by each agent at a moment, or several moments, in time.1 Aggregate analysis may have serious drawbacks. If time-series data are inadequate, a longer time series
cannot be obtained until the requisite passage of time. In this instance, estimation of
The author is assistant professor, Department of Agricultural Economics, Purdue University. Thanks are extended to Paul
V. Preckel, Wallace E. Tyner, and an anonymous reviewer for comments on earlier drafts. All errors and omissions remain
the responsibility of the author.
1Hybrid approaches which combine individual-level data and aggregate data also exist (e.g., see Stoker).
Demandfor Herbicide in Corn 205
Arndt
response parameters must proceed using cross-section data. Moreover, even if timeseries data exist, the value of such information, in the absence of supporting individuallevel data, may be limited. As Stoker points out, untenable assumptions must be
imposed on individual behavior in order to generate aggregate behavior consistent with
the theory of the consumer or the firm. Stoker states:
No realistic conditions are known which provide a conceptual foundation for ignoring
compositional heterogeneity in aggregate data, let alone a foundation for the practice
of forcing aggregate data patterns to fit the restrictions of an individual optimization
problem (p. 1829).
Stoker proposes a mixed individual data and aggregate data modeling approach. Conse-
quently, whether one is interested in individual or aggregate behavior, Stoker shows
that one must develop theoretically consistent models of individual behavior. These
models must be estimated from individual-level data.
While estimation of consumer or factor demand systems using data from individual
optimizing agents proceeds on firmer theoretical ground, a number of practical problems
present themselves (for a good overview, see Pudney). In the herbicide demand system
considered here, the problem of appropriate treatment of binding nonnegativity con-
straints (zero consumption of one or more inputs by one or more farms) is particularly
pressing. However, the issue is not unique to the data set employed in this analysis. In
estimating demand systems, one is highly unlikely to find a data set comprised of
observations from individual farms where, for all observations, all commodities in the
demand system are used in strictly positive quantities. More often, zero quantities
demanded permeate the data set.
The existence of binding nonnegativity constraints introduces vexing econometric
problems. First, the constraint censors the distribution of the error term. As a result,
the magnitude of the error cannot be estimated by traditional methods; however, it can
be confined to a specific range. If estimation proceeds by maximum likelihood, the
likelihood function must be integrated over that range. Second, theory dictates that the
prices of commodities that are not used are no longer relevant to the individual
optimizing agent. So, if a farmer uses no atrazine at price Pa, the farmer will use no
atrazine at price pa >Pa;furthermore, demand for all other commodities in the system
will remain the same (ceteris paribus). When quantity constraints bind, the reservation
price (an unobserved quantity) becomes the relevant price (Neary and Roberts). 2 A more
formal development of the role of reservation prices is included in Arndt, Liu, and
Preckel.
Estimators which treat binding constraints in demand systems estimation while
accounting for reservation prices have been proposed. Lee and Pitt (1986), and Wales
and Woodland develop dual and primal side estimators, respectively. Unfortunately,
2 The reason behind the appearance of a zero quantity demanded in a data set drives the appropriate econometric
treatment. There are other possible reasons beyond a prohibitively high market price and a resulting corner solution.
Consider two examples. First, a zero might appear in a consumer expenditure survey simply because the survey period was
too short. Second, Shonkwiler and Yen develop a model where consumers (and presumably firms) make consumption decisions
in an explicitly two-step manner. Agents first decide whether to consume a good and then, conditional on a positive
consumption decision, how much. Under the posited model, Shonkwiler and Yen show how to implement a consistent
estimator, which does not rely on reservation prices. However, for the factor demand system considered here, a corner
solution to the profit-maximization problem is the most likely explanation for the presence of zeros. Hence, reservation prices
are crucial.
206 July 1999
JournalofAgricultural and Resource Economics
severe analytical and computational difficulties limit application of the maximumlikelihood approaches of Lee and Pitt, and Wales and Woodland (van Soest, Kapteyn,
and Kooreman; Pudney; Heien and Durham; Yen and Roe). In practice, analysts often
use a modified Heckman's two-step approach suggested by Heien and Wessells.
However, Arndt, Liu, and Preckel demonstrate that, in a demand systems context, the
modified Heckman's procedure fails to account for the role of reservation prices. 3 Monte
Carlo evidence presented by Arndt, Liu, and Preckel shows that the modified Heckman's
approach performs very poorly, based on a mean square error criterion, relative to the
maximum-likelihood approach of Lee and Pitt.
In sum, analysts wishing to estimate even moderately sized demand systems effectively must choose between some modification of the Heckman's approach, which results
in estimates with very poor statistical properties, or maximum-likelihood approaches,
which are likely to be computationally complex. For larger systems with multiple
binding constraints, such as the 11-commodity consumer demand system analyzed by
Heien and Wessells, maximum-likelihood approaches are simply infeasible. 4
Here, a maximum entropy (ME) approach to demand systems estimation is employed
to estimate a demand system for herbicides from farm-level data. The approach relates
to the ME approach to ill-posed problems in production economics recently suggested
by Paris and Howitt. They state, "The challenge facing a researcher is to extract the
maximum amount of economic information from these incomplete data in a way suitable
for policy analysis" (p. 124). In this case, computational difficulties, as opposed to an illposed problem, render proper information extraction using traditional econometric
methods very difficult. The ME approach suggested here provides a means for extracting
relevant economic information from available data. Furthermore, evidence exists that
the ME approach performs well even when feasible alternatives exist. Arndt and Preckel
present Monte Carlo evidence where the ME approach performs very favorably relative
to the Lee and Pitt and modified Heckman's approaches using a root of mean square
error criterion. Finally, the approach is simple to implement. This article represents the
first application of the ME demand systems estimation approach to real data.
Mounting evidence indicates that ME approaches perform well in small samples
using a mean square error criterion (Golan, Judge, and Miller; Mittelhammer and
Cardell; Golan, Judge, and Perloff; Golan, Judge, and Karp). However, when prior information is imposed on parameter values, ME approaches also produce biased estimates
in small samples. For nonlinear econometric applications of the ME principle, such as
the demand systems application considered here, asymptotic properties are unknown.
Nonlinear ME estimators thus can be viewed as precise, but biased and potentially
inconsistent estimators in the tradition of James and Stein. In this study, it is argued
that the ME approach to demand systems estimation presented here represents a
significant step forward. Currently, available two-step estimators are practical to apply;
3
Recent findings by Shonkwiler and Yen document inconsistencies beyond the failure to treat reservation prices in the
Heien and Wessells procedure.
4
Recent innovations, such as Sobol Monte Carlo and the Gibb's sampler, have the potential to ease computational burdens. Nevertheless, integration within an optimization routine requires a high degree of accuracy. Relative to standard Monte
Carlo, "smart" Monte Carlo (such as the Gibb's sampler) can yield a reduction in the number of evaluations of the integrand
required to achieve the necessary accuracy. However, the methods must be employed for each observation requiring
integration in the data set and for each step of the optimization routine. Under these circumstances, computational burdens
quickly become significant even for very modest numbers of evaluations of the integrand, relative to standard Monte Carlo,
in the numerical integration routine.
DemandforHerbicide in Corn 207
Arndt
however, they fail to account for reservation prices and consequently perform poorly
when reservation prices matter, regardless of sample size. On the other hand, available
efficient estimators, such as the Lee and Pitt approach, are extremely difficult or impos-
sible to apply in practice.
The remainder of the article is structured as follows. First, the maximum-likelihood
approach of Lee and Pitt for treating the pervasive problem of binding quantity
constraints in farm-level data is reviewed and critiqued. 5 It is concluded that, for most
practical purposes (including the estimation problem treated here), maximum-likelihood
methods are impractical. Second, an ME estimator is presented. Finally, an application
of the ME approach to an herbicide demand system in corn, using cross-section data
from Indiana's White River Basin, is decribed. Results indicate relatively little own-
price responsiveness in herbicide use.
The Lee and Pitt
Maximum-Likelihood Approach
In the presence of zero quantities demanded, the reservation prices associated with
binding nonnegativity constraints are unobservable. They must be estimated. Lee and
Pitt (1986) proposed solving analytically for the relevant reservation price(s) and
substituting these expressions back into the system of demand equations. The method
has been applied, for example, by Lee and Pitt (1987) and by Yen and Roe. Lee and Pitt
present their estimator in general terms. For the sake of clarity and brevity, two
example regimes from the estimator actually employed in estimation are presented.
More detail on the generalized Leontief specification of the Lee and Pitt estimator is
provided in Arndt, Liu, and Preckel.
Consider a generalized Leontief profit function which applies to a sample of size N
with observations n = 1, 2,..., N:
(1)
+ zp
2p
, 1/2
profit0n ==-C OPn +
E (oi
profit
n
en N(
]),,
+ eni)Wni
1/2+
e
~i
~ P3ij(WniWnj)/
i
1
~j
i,j = 1, ... ,K,
where pn is output price, wn is a K-dimensional vector of input prices, e n is a K-dimensional vector of normally distributed random disturbances, and ai and Pijare parameters
to be estimated. In the remainder of this section, observation subscripts n will be
dropped to reduce notational clutter.
Equation (1) is a random technology specification similar to specifications employed
by Lee and Pitt (1987); Wales and Woodland; Pudney; and van Soest and Kooreman. As
Pudney points out, this specification is chosen primarily to keep the problem tractable.
As will be made clear shortly, the error structure imposed here maintains additive
error terms in the factor demand equations even after the manipulations necessary to
treat binding quantity constraints. If all behavioral parameters are viewed as random
6
For the key issues, coherency and numerical evaluation of the cumulative joint normal distribution function, the critique
of Lee and Pitt applies to the Wales and Woodland approach.
208 July 1999
JournalofAgriculturaland Resource Economics
(a and p), the manipulations necessary to treat binding quantity constraints would
generate multiplicative error terms, and thus greatly complicate estimation. Essentially, the specification provides a rationale for an additive error term in each demand
equation.
Use of Hotelling's lemma yields the following system of factor demands:
\ 1/2
(2)
xi = p 1/2iwi 1+
E
pj
i
+ eiwi-1/2l/2 > 0,
wi)
i,j=l,...,K.
For simplicity, suppose that K = 3, and that nonnegativity constraints never bind for
input three. In this system, there are four possible regimes. A regime refers to a particular permutation of binding and nonbinding constraints. In the instance considered
here, the four possible regimes are: (a) all inputs used in strictly positive quantities,
(b) input one is not used, (c) input two is not used, and (d) inputs one and two are not
used. Regimes one and two are treated in the text. Regime three is a mirror image of
regime two. Regime four is treated in Arndt, Liu, and Preckel.
Positive Demands for All Inputs
Consider an observation, n, which exhibits strictly positive demand for all inputs. Transformation of equation set (2) to eliminate heteroskedasticity yields:
1/2.,
Yi
(3)
=
XiWi1/ 2p
-1/2 =
ij
ai +
ij
P
\1/2
+ei,
= 1,...,K.
In this case, yi = xiw1p l /2 i = 1, 2, 3} has conditional trivariate normal distribution
++
with mean vector .+.
, equal to the expectation of equation set (3) and covariance matrix
+++
0
= Z = [i], where i,j = 1, 2, 3, and "+++" indicates that xi > 0 V i. One can therefore
write the probability distribution function conditional on the vector of independent
variables and parameters Pn = [wn, p, a, P, E] as:
(4)
f(Y 1,Y2,Y3
P) = (Yl1 ,Y2
,Y3;
',
Q ++)
where ( is the trivariate normal probability distribution function.
Zero Quantity Demandedfor Input One and
Positive Demands for Inputs Two and Three
Suppose that, for a given observation, quantity demanded for input one equals zero and
is strictly positive for all other inputs. The relevant price associated with input one, for
DemandforHerbicide in Corn 209
Arndt
the optimizing agent and for the purposes of estimation, is thus the reservation price
(7 1 ). The reservation price is defined as the price that would drive demand for input one
to exactly zero (Neary and Roberts). Thus, using the first equation (i = 1) of equation set
(2), one may substitute ns for wl and solve analytically for nu
1 . This manipulation gives
the reservation price in terms of the remaining market prices, the relevant parameters
of the system, and the error term as shown in equation (5) [note that w 1 2 nT
1 from
equation set (2)]:
K
1 P1/2
1/2
(5)
W
~(5)
-/
_
P S #B YW J
1+
+ E B ljW1/2
i
j=2
_
~
BL11
Bll
p1/2
e
e+
.<
P
B1 1
w
1/2
Bll
Substitution of the reservation price (71i)into the remaining equations in place of the
market price (w1) yields:
(6)
X= P
2
( W1/2
+ Bpjk.
W
|
-1/2
+ BB
jl
1/2
1/2
1/2
Equation (5) combined with equation set (6) yields an estimable system of equations
which accounts for the role of the reservation price (i7). Unfortunately, t 1, which is
unobservable, appears on the left-hand side of equation (5). However, the reservation
price (7i1 ) cannot exceed the market price (w1). Thus, the likelihood function for this
observation may be evaluated by integrating out T/2 over the interval (-oo, w /].
The variables on the left-hand side of equation (5) and equation set (6) are distributed
joint trivariate normal. The associated mean vector (f ++') and covariance matrix (Q0++)
are derived in Arndt, Liu, and Preckel. The required density function for an individual
observation in this regime takes the form:
1/2
(7)
f(x,=-,
x,
3,
P)= fw
l
4(n, X2, X,; P++, Q°-++ )d.
The function gives, for arbitrary values of the parameters (a, P, 2), the likelihood of
obtaining observed factor demands, xk {k = 1, 2, ... , K, and a reservation price (711), the
square root of which falls in the interval (-oo, w, ]. By equation (5), the root of the
reservation price is a linear function of a normally distributed random variable. Thus,
leaving aside coherency issues for the moment, the domain of integration includes
the negative quadrant of the real line (recall that one is essentially integrating out the
error term).
Critique of the Lee and PittApproach
Despite the theoretical attractiveness of the Lee and Pitt estimator, analytical and
computational difficulties severely hamper application to practical problems. On the
analytical side, the issue of coherency remains largely unresolved (van Soest, Kapteyn,
210 July 1999
JournalofAgricultural and Resource Economics
and Kooreman). Specifications of demand systems which are tractable, flexible, and
globally well-behaved are not available. For available flexible functional forms, including the generalized Leontief flexible functional form employed for this analysis, the
implications of economic theory, particularly curvature conditions, are violated for some
sets of values of parameters, exogenous variables, and error terms. Elements of these
sets are considered incoherent. If elements of these incoherent sets are not restricted
from the feasible set during maximum-likelihood estimation, van Soest, Kapteyn, and
Kooreman show that "parameter estimates will not automatically converge to values
which satisfy coherency conditions" (p. 163).
Failure to converge to values which satisfy coherency conditions can occur even if the
underlying data-generating process is coherent. Most distressingly, when parameter
estimates fail to satisfy coherency conditions, the likelihood function becomes invalid.
In particular, the sum of probabilities of events no longer equals one (van Soest,
Kapteyn, and Kooreman). Hence, maximum-likelihood estimation along the lines of
Lee and Pitt cannot be used to reject the curvature implications of economic theory on
demand systems. Failure of curvature conditions implies a violation of coherency
conditions, which in turn invalidates the estimates. On the positive side, if unconstrained estimates satisfy coherency conditions, then there is no reason to mistrust
those estimates.
When confronted with an incoherent set of estimates, the obvious solution is to
restrict the feasible space for parameter estimates and error terms. Unfortunately,
necessary and sufficient conditions, which guarantee coherency, are extremely difficult
to derive in general. For some very simple models, "almost necessary" conditions have
been derived (van Soest and Kooreman; Lee and Pitt 1987). However, van Soest,
Kapteyn, and Kooreman assert that, in practice, coherency conditions are likely to be
very restrictive and may even prevent the model from being capable of explaining the
data (i.e., estimation is infeasible). In addition, imposition of coherency conditions on
error terms would add more complex truncations to an already complicated likelihood
function. Finally, even if an acceptable solution to the coherency issue could be found,
the need to evaluate the cumulative joint normal distribution function, with a fully general covariance matrix, limits application to demand systems with a maximum of three
(perhaps four) binding constraints in a single observation (Arndt, Liu, and Preckel;
Pudney; Breslaw).
In sum, application of the Lee and Pitt estimator is severely constrained by analytical and computational difficulties. For the herbicide demand system considered
here, the limited scope of the demand system (only three goods) keeps dimensions
of integration within feasible limits. Unfortunately, unconstrained estimation via the
Lee and Pitt approach yields estimates which fail to satisfy coherency conditions.
Consequently, the likelihood function is invalid. Imposition of coherency conditions
would be extremely difficult. It would require, among other items, restrictions upon
feasible values for the error terms. As mentioned above, this implies further complex
truncations of the normal distribution. Finally, since feasible coherency conditions are
sufficient but not necessary, development and imposition of coherency conditions might
generate a conflict between the need to satisfy (sufficient) coherency conditions and the
need to explain the data. In view of this unhappy situation, it seems wise to explore an
alternative approach.
Arndt
Demandfor Herbicide in Corn 211
Maximum Entropy Estimation with
Endogenous Reservation Prices
The maximum entropy approach is motivated by the work of Shannon, who defined a
function to measure the uncertainty, or entropy, of a collection of events, and by Jaynes
(1957a, b), who proposed maximizing that function subject to appropriate consistency relations, such as moment conditions, and adding-up constraints. The maximum
entropy principle and its sister formulation, minimum cross-entropy (MCE), is now used
in a wide variety of fields to make inferences when information is incomplete, highly
scattered, and/or inconsistent (Kapur and Kesavan). Outside of economics, ME and MCE
have been employed in pattern recognition, spectral analysis, and queuing theory, to
name a few examples. In economics, MCE is currently widely employed in general
equilibrium applications to balance social accounting matrices (Golan, Judge, and
Robinson).
Recently, Golan, Judge, and, Miller developed a reformulation of the general linear
econometric model in order to permit parameter estimation by the entropy principle.
The entropy principle has been successfully applied to a range of econometric problems,
including nonlinear problems, where limited data and/or computational complexity
hinder traditional estimation approaches. For example, Golan, Judge, and Karp apply
the maximum entropy approach to dynamic estimation problems where the state
variables are unobserved ("counting fish in the sea"). Golan, Judge, and Perloff apply
the maximum entropy approach to censored regression (Tobit) and ordered multinomial
response models. In both studies, Monte Carlo experiments are presented which, using
a mean square error criterion, illustrate very favorable performance of the ME approach
compared with maximum likelihood under a wide range of conditions. Performance of
the ME approach, relative to maximum likelihood, is particularly impressive in the
presence of nonnormally distributed error terms, multicollinearity, or other departures
from ideal estimation conditions. Finally, maximum entropy approaches have been
applied to estimation ofhousehold models (Heltberg, Arndt, and Sekhar) and production
technologies (Paris and Howitt).
In all of these applications, the value of entropy or cross-entropy is a distance metric
from a prior distribution. If the prior distribution is uniform, maximum entropy is a
measure of information content in the constraints, since the uniform is considered the
most uncertain, or least informative, distribution. In ME econometric applications of the
form proposed by Golan, Judge, and Miller, prior distributions are usually imposed on
both the parameters to be estimated and the error terms. Thus, parameter estimates
must fall within a specified range of values (which can be very large) and estimated
error terms can be no larger in absolute value than some prespecified bound. The
optimal solution reflects tension between choosing parameter values, which allow the
model to closely fit the data (in an entropy metric sense), and parameter values, which
are close (also in an entropy metric sense) to prior parameter values. In choosing the
prior distributions, the analyst implicitly chooses the relative weight between the dual
objectives of fitting the data and of remaining close to prior values for parameters.
For the general linear model, asymptotic properties of the estimates and concomitant
statistical tests have been developed (Mittelhammer and Cardell). In addition, Golan,
and Golan and Judge present an entropy ratio statistic analogous to a likelihood ratio.
The statistic appeals to information theory. It permits hypothesis testing by measuring
212 July 1999
Journalof Agriculturaland Resource Economics
the information content of additional constraints. This statistic is discussed in the
appendix and is employed for hypothesis testing in the application.
The ME approach employed here generally follows that of Golan, Judge, and Miller.
The estimator departs from the approach of Golan, Judge, and Miller in two ways. First,
prior distributions are imposed on elasticity estimates rather than parameter values.
Elasticity priors have the advantage of being unit free and more transparent (especially
for the generalized Leontief functional form presented here). Second, reservation prices
associated with binding constraints are specified as variables.
For convenience, equation set (3) is rewritten as:
(8)
~~(8)
V
Yi
=
~Yi
i-p1/2w
ZViP
X
1 /2
W-i
+ =
ai +
Ili
E ,J()
e
e.
+ ei
2
0,
O.
i,j 1,...,K,
where xj equals the market price (wj) if yj > 0, and nj equals the reservation price if
yj = 0. This system of equations is estimable via the maximum entropy criterion. For the
cases whereyi = 0,
Oi, the associated reservation price, is treated as a variable. For given
estimates of the parameters and error terms, estimates of the reservation prices ('Ai,
where yi = 0) can be determined via the same set of analytical manipulations employed
by Lee and Pitt. Market prices for commodities that are not used (wi) serve as an upper
limit for the variables representing reservation prices (7i,where yi = 0). The model can
be represented as:
(9)
Y = HY + e,
where Y is an {N x K vector of observations; II is an {N x (K + 1)} matrix containing
square roots of normalized market prices, endogenously determined square roots of normalized reservation prices, and a vector of ones; y is a {(K + 1) x K } matrix containing
the unknown parameters a and P; and e is an {N x K} matrix of random error terms.
Define %i as the elasticity of input i with respect to pricej. Reparameterization of this
model proceeds as follows. Treat each Ei as the expectation of a discrete random variable
with compact support and 2 < M < oo possible outcomes. Thus, if M = 2, and zijl and Zij2
are plausible upper and lower bounds, ei can be expressed as:
(10)
ei = rij+
(1- r)ij,
where ri is the probability of outcome z^i. Similarly, treat eni as the expectation of a
finite and discrete random variable with 2 <J < oo possible outcomes. Thus, if J =2, and
v,, and Vn, are plausible upper and lower bounds of e, one can express en as:
(11)
eni = qninil + (1 -
ni)ni2
where qn is the probability of outcome v,.l.
Each equation in the reparameterized system can be written in matrix form as:
(12)
Yi = IIi + ei = IIy, + Viqi,
Demandfor Herbicide in Corn 213
Arndt
where y, represents column i of the matrix y. The matrix on the far right-hand side of
(12) takes the following form:
e i = Viq
(13)
i
vUi
0
0
vi 2
0
0
=
0
qi,
· 0
qi 2
.
viN
qiN
where Vi is an {N x NJ } matrix of error support points, and qi is an {NJ x 1 } vector of
probabilities.
In the maximum entropy approach presented here, parameter estimates are obtained
by maximizing the following constrained optimization problem:
N
K K+1 M
(14)
Max
-E
s.t.:
J
Yi = IIYi
+
)
n=l i=l j=l
i=l j=l m=l
r,q,T,y,e
K
rijln(rij ) - E E E qnijln(qi
Viqi
V i =1,...,K,
M
=
m=l
i,j=1 ..
rijmZijm
i=
Eii = fi(y,P'W)
Ei= hl(Y,
,W)
, K,
,...,K,
Vi,j = 1,..., K,
i #j,
M
1 = E rijm
Vi=1,...,K,
j = 1,...,K,
m=l
J
1
= qnij
j=1
nni
7ni
Vi=1,...,K,
> 0
=
Wni if Yni
<
Wni if Yi = 0
n=1,...,N,
i = 1,...,K,
n =,...,
i
n= 1,...,N,
1,...,K,
where fi and hiy are, respectively, standard own- and cross-price elasticity calculations
for the generalized Leontief functional form, and bars over prices p and w indicate
7
average prices.6 Other restrictions, such as symmetry conditions, are easily imposed.
Extension to a cross-entropy approach is straightforward. Note that no prior distributions are imposed for the reservation prices (7ni, where Yni = 0). As shown in the last
restriction of (14), these are variables which are constrained to be less than or equal to
the market price (wni).
the limit, Oln(O) = 0. In practice, very small values are imposed as lower bounds on probabilities r and q.
2
For maximum entropy estimation of the general linear model, measures of fit (pseudo-R ) are available (Golan, Judge,
and Miller). For this nonlinear application with censored data, the concept of "explained variation" becomes more slippery.
Hence, measures of goodness of fit are not presented.
6 In
7
214 July 1999
JournalofAgricultural and Resource Economics
Comparison of the ME approach suggested in (14) with direct maximum-likelihood
estimation seems relevant at this point. Four items bear particular mention. First, in
stark contrast to maximum-likelihood formulations, the ME approach is easy to set up
and solve. Since the maximization problem in (14) does not involve numerical integration, it usually can be solved using standard nonlinear optimization packages such as
GAMS (Brooke, Kendrick, and Meeraus). Second, while the computational advantages
of the ME approach are clear, the implications of failure of coherency conditions under
ME estimation are substantially less clear. Van Soest, Kapteyn, and Kooreman are
careful to confine their discussion of coherency issues to maximum-likelihood estimation. That failure of coherency conditions might invalidate other estimators, such as the
ME approach, is neither asserted nor precluded. This is an important topic for further
research. At the moment, it appears we can only state that failure to atisfy coherency
conditions may or may not invalidate the ME estimator.
Third, should one choose to impose coherency conditions, the ME approach facilitates
imposition of these conditions. In particular, relative to maximum likelihood, inequality
constraints on values for error terms are quite simple to impose. Fourth, the ME
approach suggested in (14) shares the advantages of the general ME approach. The
distributions on parameter values allow one to impose prior information. If, from theory
or through previous work, the analyst has information on likely values or sign conditions for parameters, this information can be easily applied through simple inequality
constraints, the choice of prior distributions for parameters, and/or a cross-entropy
formulation (Golan, Judge, and Miller).
Data, Estimation, and Results
Here, the responsiveness of corn growers in Indiana's White River Basin to changes in
herbicide prices is examined. A detailed description of the data and region can be found
in Rudstrom. Briefly, the White River Basin covers an area of about 11,000 square miles
in south central Indiana. Topography and soil characteristics vary from flat, highly
productive soils found in the northern third to steeply sloped (> 6%), less productive
soils in the south central portion. Corn and soybeans dominate cropping patterns,
though some wheat and sorghum are grown as well.
This study focuses on chemical weed control options in corn production. In Indiana
in 1993, nearly all of the area planted to corn received some herbicide application. In the
White River Basin, herbicides are typically applied as pre-plant incorporate or in a preemergent (after planting but prior to emergence of the seedling) application. Quantity
of herbicide applied can be modulated by choice of application method. In a blanket
application, the full field receives the same dose. Band applications concentrate doses
in the row near the crop. In a spot application, the farmer concentrates doses in areas
where weeds tend to be the most problematic.
The data set includes information on price and quantity applied per acre (pre-plant
incorporate and pre-emergent) for three herbicides [atrazine, cyanazine, and a composite
of metolachlor and alachlor (M-A) corresponding to variables x, x 2, and x 3, respectively],
as well as information on corn price from 223 farms for the 1992-93 cropping season.
Zero quantities demanded permeate the data set. As shown in table 1, only three fields
are treated with all three herbicides simultaneously. Atrazine is nearly always applied
Arndt
Demandfor Herbicidein Corn 215
Table 1. Importance of Different Regimes
Regime
a
No. of
Observations
Percent of
Observations
Atrazine
Cyanazine
M-A b
+
+
+
31
0
+
+
1
0
+
+
0
+
+
0
121
19
54
0
0
+
11
5
0
+
0
1
0
+
0
0
67
30
223
100
Total:
8
aA "+" indicates positive consumption of the input, while a "O"indicates zero quantity demanded.
b M-A is a composite of metolachlor and alachlor.
cPercentages do not total to 100 due to rounding errors.
(only 13 fields receive no atrazine), while cyanazine is only applied to 24 fields. The M-A
composite is applied to about 60% of the fields in the sample.
Unfortunately, the data set does not contain information on tillage-a substitute for
herbicides. In ordinary least squares (OLS) estimation, omission of important variables
usually results in biased estimates. Bias stems from correlations between included and
omitted important variables. In OLS, direction of bias also may be ascertained. Positive
(negative) correlation between observed independent variables (X) and the error term
(e) results in positively (negatively) biased parameter estimates. Though the ME esti-
mator employed here is more complex than OLS, it is likely to have the same difficulty
in distinguishing between effects of independent variables and the error term when
correlations exist between the independent variables and the error term. 8 Thus, when
interpreting regression results, the possibility of bias due to omitted variables must be
kept in mind.
In the ME formulation, results can be sensitive to the prior distributions specified for
elasticities and error terms, Z and V, respectively (Mittelhammer and Cardell). Prior
elasticity values are presented in table 2. Weights on all prior elasticity points are equal.
Since theory indicates that own-price elasticities should be negative, prior distributions
on elasticities are skewed negatively. These prior distributions permit own-price elasticity values to range from mildly positive (and counter to the predictions of theory) to
highly responsive. The distributions are meant to strictly contain the plausible set of
elasticity values.
For cross-price elasticities, theory does not imply sign conditions, and little prior
information is available. For these elasticities, prior distributions are set quite wide
using equally weighted three-point prior distributions with points {-1, 0, 1}. For the
8 Mittelhammer and Cardell find the ME approach to be more robust to misspecification relative to OLS for the general
linear model.
JournalofAgricultural and Resource Economics
216 July 1999
Table 2. Elasticity Prior Points for the Herbicide
Demand System
Prior Points
Elasticity
c.··
~
eu
Low
Center
High
-2.0
-0.8
0.4
-1.0
0.0
1.0
Note: i,j = 1, 2, 3 (1 = atrazine, 2 = cyanazine, 3 = M-A composite).
Table 3. Data Points and Own-Price Elasticity Estimates
Herbicide
Atrazine
Cyanazine
M-A
Price ($/unit)
3.19
6.31
7.08
Quantity (units/acre)
1.45
1.96
1.91
-0.50
(0.62)
-0.02
(0.01)
-0.05
(0.04)
-0.71
(infeasible)
-0.06
(0.27)
-0.17*
(3.42)
Description
Est'd. Elasticity:
Unconstrained caseb
Constrained case
Notes: Data evaluated at mean prices for all observations and mean quantities for all nonzero observations. An asterisk (*) denotes rejection of the null hypothesis of zero elasticity at the 90% confidence level.
Numbers in parentheses below the estimates are the entropy ratio statistic on the test for a zero own-price
elasticity. The critical value at the 90% confidence level is 2.706.
aM-A is a composite of metolachlor and alachlor.
b No coherency conditions are imposed.
Coherency conditions are imposed.
error terms, equally weighted two-point prior distributions are employed with points
{-9, 9}. This choice corresponds to very wide error bounds, approximately eight error
term standard deviations, as recommended by Preckel. In accordance with theory,
symmetry of the P matrix was imposed. In addition, reservation price estimates were
constrained to be nonnegative and less than the prevailing market price. Estimation
was performed in GAMS.
Regression results are presented in table 3. Estimates without coherency conditions imposed violate curvature conditions for some observations. Consequently, a choice
must be made. One can either accept the incoherent estimates which might or might
not be invalid, or one can impose sufficient coherency conditions which might or might
not be overly restrictive. Results from both cases are presented. Sufficient coherency
conditions are imposed by adding the following constraints to the optimization problem
in (14):
Demandfor Herbicide in Corn 217
Arndt
Pj >0
(15)
ai
+
e ni
>
0
i,j=l,...,K, i j,
Vi=l,...,K,
n=l,...,N.
These constraints ensure global convexity of the profit function for each observation;
however, they also preclude complementarity between inputs (Chambers; Terrell). 9 The
estimation scheme without the equations in (15) imposed is labeled the unconstrained
case, while the estimation scheme with the equations in (15) imposed is labeled the
constrained case.
For the unconstrained case, estimated cross-price elasticity values are small and
strictly positive. Sensitivity analysis with respect to the prior distributions for this case
revealed that signs on the cross-price effects are sensitive to the specification of prior
distributions on parameters. For the constrained case, estimated cross-price elasticities
are all zero (suggesting that the constrained formulation is in fact overly restrictive).
With the exception of atrazine, own-price elasticity estimates tended to be relatively
robust for both cases; consequently, the results tables concentrate on own-price elas-
ticity values.
Own-price elasticity estimates (presented in table 3) reflect the standard elasticity
calculation imposed in the constraint set of (14) using average prices over all observations and average quantities demanded over all consuming observations.1 0 They are
qualitatively similar for the two cases. The results indicate relatively little sensitivity
of herbicide use in corn to changes in own price. Tests using the entropy ratio statistic
find none of the estimated elasticities to be significantly different from zero for the
unconstrained case. In the constrained case, the own-price elasticity estimate for M-A
is significantly different from zero at the 90% confidence level. In addition, imposing a
zero own-price elasticity for atrazine is infeasible. 1l
Our interest here is to examine the likely incidence of taxes on herbicides. Recall that
inelastic factor demands imply greater incidence on farmers. The direction of shift from
the prior own-price elasticity value (-0.8) provides some information regarding the
estimated elastic is lower than
likely magnitude of the elasticities. For all inputs, thee essity
the (inelastic) value implied by the parameter prior distribution. To further examine the
incidence issue, a test was conducted to determine if the data rejected a hypothesized
own-price elasticity of -1.5 for each herbicide. Tests were conducted for each herbicide
separately and for the three herbicides as a group. Results are presented in table 4. In
the unconstrained case, the null hypothesis of elastic own-price responsiveness for
cyanazine and M-A is rejected at the 90% confidence level. The test fails to reject the
null hypothesis for atrazine. The data reject the hypothesis of elastic own-price
responsiveness for all herbicides simultaneously at the 95% confidence level. In the
constrained case, the data fail to reject the hypothesis of elastic response for atrazine
9 Terrell advances a Bayesian approach to ensuring coherency conditions over a range of the data. Application of the
approach of Terrell to demand systems with multiple regimes might be a fruitful avenue for future research. The author
thanks an anonymous reviewer for pointing out this reference.
10The presence of multiple regimes and the censoring of the error term lead to a multiplicity of possible elasticity calculations. Greene provides a review of the debate over the appropriate elasticity measure. Analysis of elasticities by regime
yielded similar qualitative results.
n Infeasible may be taken as a rejection of the null hypothesis since it indicates that the data are completely incompatible
with the hypothesis. It is important to note that, since the constraints are nonlinear, it is possible that a feasible solution
exists and the optimization routine could not find it.
JournalofAgricultural and Resource Economics
218 July 1999
Table 4. Tests for Elastic Own-Price Response
Herbicide
Atrazine
Cyanazine
M-A a
Group b
Unconstrained case
1.02
3.80*
3.28*
8.43**
Constrained cased
0.82
infeasible
1.72
Description
Entropy Ratio:
infeasible
Notes: Single and double asterisks (*) denote rejection of the null hypothesis at the 90% and 95% confidence levels, respectively. The critical value for the individual tests at 90% confidence is 2.706; the critical
value for the group test at 95% confidence is 7.815.
a M-A is a composite of metolachlor and alachlor.
b Imposition of elastic response (Ei = - 1.5) for all herbicides simultaneously.
cNo coherency conditions are imposed.
d Coherency conditions are imposed.
and M-A. For cyanazine and the three herbicides as a group, imposition of elastic
response is infeasible.
Overall, information gleaned from the data tentatively paints a picture of limited
own-price response (inelastic factor demand) for the three herbicides considered. This
conclusion is most tentative for atrazine, which exhibited the greatest sensitivity of
specification of prior elasticity bounds, the highest estimated elasticity, and failure to
reject the hypothesis of elastic response for both the constrained and unconstrained
cases. Nevertheless, on balance, the results suggest that taxes on herbicides would have
relatively little impact on quantities of herbicide demanded by individual farms in the
White River Basin. Thus, the results indicate that the incidence of an herbicide tax
would fall primarily on farmers.
Summary and Conclusions
Due to the impacts of aggregation across heterogeneous optimizing agents, estimation
using micro-level data has attractive theoretical properties. Yet, practical econometric
problems hinder empirical work. Here, focus is on one practical problem-binding nonnegativity constraints. The problem is pervasive and results in vexing econometric
difficulties. In particular, theory indicates that unobserved reservation prices (not
market prices) for commodities that are not used are the relevant prices to the optimizing agent. Maximum-likelihood approaches, which account for the role of reservation
prices, exist (Lee and Pitt 1986; Wales and Woodland); however, application of these
approaches is severely limited by analytical and computational difficulties.
In the herbicide demand system considered here, the Lee and Pitt approach was
applied first. However, unconstrained estimation via the Lee and Pitt approach resulted
in parameter estimates which failed to satisfy coherency conditions. Consequently,
the Lee and Pitt estimates are invalid. The difficulties associated with imposing coherency conditions onto an already complex estimator led to a search for an alternative
approach.
Arndt
DemandforHerbicide in Corn 219
The alternative employed here is a maximum entropy (ME) approach. The maximum
entropy principle has been successfully applied to a number of estimation problems
where computational complexity and/or limited data hinder traditional estimation
approaches. Monte Carlo evidence indicates that the ME approach performs very well
relative to the Lee and Pitt and modified Heckman's approaches using a mean square
error criterion (Arndt and Preckel). In addition, relative to maximum-likelihood
approaches, the ME approach presented here is simple to implement, particularly if
coherency conditions must be imposed on error terms. Finally, the approach shares the
general advantages associated with ME estimation.
The approach was applied to an herbicide factor demand system for corn growers in
Indiana's White River Basin. The results obtained indicate relatively little sensitivity
of herbicide demand to changes in own price. This implies that the incidence of taxes on
herbicides would fall primarily on farmers.
Data omissions, incoherencies, restrictive model formulations, failure to reject some
hypothesis tests, and sensitivity of results to prior distributions, particulary for
atrazine, render these conclusions tentative. In general, for the cases where reservation
prices matter, the ME approach represents a step forward from the current situation
where available estimators which are practical to apply have poor statistical properties,
and available consistent estimators are extremely difficult or impossible to apply in
practice. Based on experience to date, the maximum entropy approach to demand
systems estimation appears to merit further attention.
[Received June 1998;final revision receivedFebruary 1999.]
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DemandforHerbicide in Corn 221
Arndt
Appendix:
The Entropy Ratio Statistic
Denote zu as the objective value for the optimization problem in (14) unencumbered by any hypothesis
test, and denote zc as the objective value for the optimization problem in (14) when a constraining
hypothesis (such as the own-price elasticity for cyanazine equal to -1.5) has been added to the
constraint set. The test statistic, X,is then:
X = 2z(1
C
which converges in distribution to Xk with k degrees of freedom in large samples. Degrees of freedom
correspond to the number of constraints imposed.
As noted in the introduction, the entropy objective is a measure of information content of the
constraints when flat priors are imposed (as is the case in this analysis). Flat priors represent the
greatest entropy or least information. When an arbitrary constraint is imposed, such as own-price
elasticity of cyanazine equal to -1.5, a large reduction in the objective value implies that the constraint
is highly informative. In other words, the constraint adds significant information beyond the
information content derived from the data. In these cases, the null hypothesis represented by the
constraint is rejected.